Take X and Y to be the set of real numbers
This subset specifies the
relation y >= x
Y
x=y
X
This subset specifies the relation x2 + y2 <=1
Y
x2 + y2 =1
X
Relation Tables
Student Info
Name
UF_ID
Major
GPA
John
1111-2221
Math
3.9
Mary
2222-1111
CS
3.2
Helen
2333-5555
CS
3.9
Jean
5555-1234
Chemistry
3.8
A simple example of a database
Columns are attributes and each row is an element in a
Cartesian product AxBxCxD (what are them?)
Equivalence Relation
A relation is an equivalence relation if it is reflexive,
symmetric and transitive.
= is an equivalent relation
> and < are not equivalence relation
The relation (a, b and n are integers)
a Ξ b (mod n) is an equivalence relation.
a Ξ b (mod n) reads “a is congruent to b module n” (or a =
b mod n).
a Ξ b (mod n) if a- b divides n or a, b have the same
remainder upon dividing by n.
It is reflexive, symmetric and transitive.
a Ξ a (mod n)
a Ξ b (mod n) -> b Ξ a (mod n)
a Ξ b (mod n) and b Ξ c (mod n) ->
a Ξ c (mod n)
Example
R={ {1, 1}, {2, 1}, {3, 2}, {4, 3} }
R2 = { {1, 1}, {2, 1}, {3, 1}, {4, 2} }
R3 = { {1, 1}, {2, 1}, {3, 1}, {4, 1} }
R4 = ?
U
Theorem: The relation R is transitive if and only if
Rn
R for all n=1, 2, 3, …
Proof: “if” part. (a, b), (b, c) are in R, then (a, c) is in R
also.
“only if” use induction.
Suppose Rn is in R for n < = k
Rn + 1 = Rn o R will be in R o R which is in R.